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Chapter 4: Problem 10

\(\frac{5}{12} \times \frac{2}{3}+1 \frac{7}{8}\) (a) \(3 \frac{7}{24}\) (b) \(4 \frac{3}{8}\) (c) \(2 \frac{11}{72}\) (d) \(1 \frac{5}{6}\)

Short Answer

Expert verified
The answer is (c) \(2 \frac{11}{72}\).

Step by step solution

01

Multiply the fractions

Start by multiplying the fractions \( \frac{5}{12} \) and \( \frac{2}{3} \). To do this, multiply the numerators together and the denominators together: \( \frac{5 \times 2}{12 \times 3} = \frac{10}{36} \). Simplify \( \frac{10}{36} \) by dividing both the numerator and the denominator by 2, the greatest common divisor, giving \( \frac{5}{18} \).
02

Convert the mixed number to an improper fraction

The mixed number \( 1 \frac{7}{8} \) needs to be converted to an improper fraction. Multiply the whole number by the denominator and add the numerator: \( 1 \times 8 + 7 = 15 \). So, \( 1 \frac{7}{8} = \frac{15}{8} \).
03

Add the two fractions

To add \( \frac{5}{18} \) and \( \frac{15}{8} \), find a common denominator. The least common multiple of 18 and 8 is 72. Convert \( \frac{5}{18} \) to \( \frac{20}{72} \) and \( \frac{15}{8} \) to \( \frac{135}{72} \). Now, add the fractions: \( \frac{20}{72} + \frac{135}{72} = \frac{155}{72} \).
04

Simplify the result

Convert \( \frac{155}{72} \) to a mixed number by dividing 155 by 72. The quotient is 2 and the remainder is 11, which gives \( 2 \frac{11}{72} \). Check that no further simplification is possible for the fraction part. Hence, the final answer is \( 2 \frac{11}{72} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Multiplication
Multiplying fractions involves a simple yet crucial process. You start with two fractions, let's say \( \frac{5}{12} \) and \( \frac{2}{3} \). The way to multiply them is to handle the numerators and denominators separately.
You multiply the numerators (top numbers) together and the denominators (bottom numbers) together like this:
  • Numerator: \( 5 \times 2 = 10 \)
  • Denominator: \( 12 \times 3 = 36 \)
This gives you the new fraction \( \frac{10}{36} \).
To make it neat, you should simplify the fraction. Look for a number that divides both the numerator and the denominator. Here, you can use 2:
  • Divide both by 2: \( \frac{10}{36} \rightarrow \frac{5}{18} \)
Now, your fraction is as simple as it can get, ready to be used in further calculations.
Mixed Numbers
Mixed numbers combine whole numbers and fractions. They look like this: \( 1 \frac{7}{8} \). You see a whole number '1' and a fraction '7/8'.
To make calculations easier, it's often useful to convert mixed numbers into improper fractions. This involves:
  • Multiplying the whole number by the fraction's denominator: \( 1 \times 8 = 8 \)
  • Adding the numerator: \( 8 + 7 = 15 \)
So, \( 1 \frac{7}{8} \) becomes \( \frac{15}{8} \). In this form, it's more straightforward to add or multiply these fractions.
Improper Fractions
Improper fractions have numerators larger than their denominators, for example, \( \frac{15}{8} \). Although they may look cumbersome, they are very handy in calculations.
One good thing about improper fractions is that they can easily be converted back into mixed numbers:
  • Divide the numerator by the denominator: \( 15 \div 8 \)
  • The quotient is the whole number, and the remainder forms the fractional part.
In our example, \( 15 \div 8 = 1 \) R7, so you have \( 1 \frac{7}{8} \) again. The mixed number is often more intuitive to grasp, but improper fractions are mathematically more flexible.
Addition of Fractions
Adding fractions requires a shared baseline: the common denominator. It's crucial to find one, especially when fractions have different denominators, like \( \frac{5}{18} \) and \( \frac{15}{8} \).
Here's the step-by-step guide:
  • Find the least common multiple (LCM) of the denominators. For 18 and 8, the LCM is 72.
  • Convert each fraction to have this denominator.
  • \( \frac{5}{18} \) becomes \( \frac{20}{72} \) and \( \frac{15}{8} \) becomes \( \frac{135}{72} \).
Now, they can be added:
  • Add the numerators: \( 20 + 135 = 155 \)
  • Keep the common denominator: \( \frac{155}{72} \)
Converting this sum into a mixed number gives the final answer, \( 2 \frac{11}{72} \). Therefore, finding a common denominator simplifies the process, leading to a clean and correct result.

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Most popular questions from this chapter

Find the cost of \(2 \frac{1}{4}\) lbs of bananas at \(60 \notin\) per pound. (a) 125垄 (b) 145垄 (c) 155垄 (d) 135垄

Divide \(\frac{1}{6} \div \frac{2}{9}\) (a) \(\frac{2}{53}\) (b) \(1 \frac{1}{9}\) (c) \(\frac{1}{27}\) (d) \(\frac{3}{4}\)

Subtract \(-1 \frac{1}{8}-5 \frac{2}{3}\) (a) \(4 \frac{13}{24}\) (b) \(6 \frac{19}{24}\) (c) \(-6 \frac{19}{24}\) (d) \(-4 \frac{13}{24}\)

$$\operatorname{Add} \frac{11}{12}+\frac{5}{8}$$ $$\begin{aligned} &\text { (a) } 1 \frac{5}{6}\\\ &\text { (b) } 1 \frac{3}{4} \end{aligned}$$ (c) \(1 \frac{13}{24}\) (d) \(2 \frac{1}{8}\)

Which fraction is the smallest: \(\frac{5}{16}, \frac{3}{8}, \frac{1}{4}, \frac{1}{3} ?\) (a) \(\frac{5}{16}\) (b) \(\frac{3}{8}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{3}\)
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